# Do we keep the instrument in the OLS, if Wu-Hausman test (using that instrument) fails to reject that OLS is consistent?

If two instrument variables (for example $Z_1$ and $Z_2$) are both exogenous and relevant. And we found Wu-Hausman test fails to reject the $H_0$: Both IV regression and OLS are consistent. When we want to report the OLS result:

We have two options: $$(1) Y = X_1 + X_2 + ... + Z_1 + Z_2 + e$$ $$(2) Y = X_1 + X_2 + ... + e$$ Is there any compulsory reasons that $Z_1 + Z_2$ need to be in the equation? Thanks very much!

Look at both the OLS and the IV estimates, and in particular, look at their precisions. If they are imprecise, that might explain why the test does not reject the null. If both are very precise (and close to each other), then you have no reason not to report it, because it is an interesting result. In general, my advice would be to report both the OLS (without including $Z_1$ and $Z_2$ among the regressors) and the IV estimate results (as well as the result of the Wu-Hausman test).
• Hi Ray, Thanks for clarifying the question, I understand better now. The usual practice would be not to include $Z_1$ and $Z_2$ in the OLS regression. I'll edit the answer to make this clear. Sep 13, 2018 at 22:21